LINES AND ANGLES 99
Now, measure any pair of corresponding angles and find out the relation between
them. You may find that : ✁1 = ✁5, ✁2 = ✁6, ✁4 = ✁8 and ✁3 = ✁7. From this,
you may conclude the following axiom.
Axiom 6.3 : If a transversal intersects two parallel lines, then each pair of
corresponding angles is equal.
Axiom 6.3 is also referred to as the corresponding angles axiom. Now, let us
discuss the converse of this axiom which is as follows:
If a transversal intersects two lines such that a pair of corresponding angles is
equal, then the two lines are parallel.
Does this statement hold true? It can be verified as follows: Draw a line AD and
mark points B and C on it. At B and C, construct ✁ABQ and ✁BCS equal to each
other as shown in Fig. 6.20 (i).
Fig. 6.20Produce QB and SC on the other side of AD to form two lines PQ and RS
[see Fig. 6.20 (ii)]. You may observe that the two lines do not intersect each other. You
may also draw common perpendiculars to the two lines PQ and RS at different points
and measure their lengths. You will find it the same everywhere. So, you may conclude
that the lines are parallel. Therefore, the converse of corresponding angles axiom is
also true. So, we have the following axiom:
Axiom 6.4 : If a transversal intersects two lines such that a pair of corresponding
angles is equal, then the two lines are parallel to each other.
Can we use corresponding angles axiom to find
out the relation between the alternate interior angles
when a transversal intersects two parallel lines? In
Fig. 6.21, transveral PS intersects parallel lines AB
and CD at points Q and R respectively.
Is ✁BQR = ✁QRC and ✁AQR = ✁QRD?
You know that ✁PQA = ✁QRC (1)
(Corresponding angles axiom)
Fig. 6.21